Homework 07: QMC for harmonic bosons
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Introduction

In this exercise, as in Class Session 07 , we consider N non-interacting bosons in an harmonic trap. We focus on the energy of those bosons using a simple direct-sampling Quantum Monte Carlo algorithm.

Thermodynamics of N non-interacting bosons

We take the notations of Class Session 07.

A- The energy of non-interacting bosonic particles

• -1- Consider the following expression for the mean energy ⟨E⟩ at inverse temperature β

$\langle E\rangle = -\frac 1{Z_N} \frac{\partial Z_N}{\partial\beta}$

• From the recursion relation (1) of Class Session 07, write the mean energy ⟨E⟩ in terms of the partition functions Zn and zn and their derivatives with respect to β, ∂Zn/∂β and ∂zn/∂β.

• -2- We now chose a harmonic trap the such that the energy levels are En = n in each of the three spatial directions. The (three-dimensional) single particle partition zk writes:

$z_k= \Big(\frac{1}{1-e^{-k\beta}}\Big)^3$

• From this expression, find a relation between ∂zk/∂β and zk.

• -3- Inspired by the algorithms of Class Session 07, write a program to compute the mean energy by using a recursion relation on the pair (ZN,∂ZN/∂β). Plot the mean energy as a function of the reduced temperature T٭ for different values of N. Consider a sufficiently large range of reduced temperatures. Comment on the behavior with N. Identify and give an explanation for the the high-temperature asymptotics. In particular, what is the advantage of defining T٭, where does the N1/3 of its definition come from?

Important remark: at fixed N and high temperature, you may obtain irrelevent results. This is due to an overflow in the computation of the Zn's. You either have to check that no overflow occurs, or, better, find a cure by changing the normalizations of the partition functions.

B- The condensate fraction of non-interacting bosonic particles

• -1- Consider the partition function W≥k of N bosons with ≥k of them in the ground state (energy 0). Show that

$W_{\geq k}= Z_{N-k}$

• -2- Consider the partition function Wk of N bosons with precisely k of them in the ground state. Show in details that

$W_{k}= \left\{\begin{array}{ll}W_{\geq k}-W_{\geq k+1} & \text{if}\quad k<N\\ W_{\geq k}& \text{if}\quad k=N\end{array}\right.$

• -3- Deduce from those results the probability π(N0) of having N0 bosons in the ground state in terms of the partition function.
• -4- The condensate fraction, which is the mean value ⟨N0⟩ of the number N0 of bosons in the ground state, writes

$\langle N_0\rangle = \sum_{N_0=0}^N N_0 \pi(N_0)$

• Using the results of the previous questions, show that

$\langle N_0\rangle = \frac 1{Z_N}\sum_{p=0}^{N-1} Z_p$

• -5- Modify the program you've written in the previous section so as to include a computation of ⟨N0⟩. Plot this quantity as a function of the reduced temperature T٭ for different increasing values of N. Comment.

References

• Borrmann P., Franke G. (1993) Recursion formulas for quantum statistical partition functions, Journal of Chemical Physics 98, 2484–2485
• Landsberg P. T. (1961) Thermodynamics with quantum statistical illustrations, Interscience Publishers
• SMAC part 4.2.3 - 4.2.6

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